Lemmas about List.minimum? and `List.maximum?.

Minima and maxima

minimum?

@[simp]

theorem List.minimum?_nil {α : Type u_1} [Min α] : [].minimum? = none

theorem List.minimum?_cons {α : Type u_1} {x : α} [Min α] {xs : List α} : (x :: xs).minimum? = some (List.foldl min x xs)

@[simp]

theorem List.minimum?_eq_none_iff {α : Type u_1} {xs : List α} [Min α] : xs.minimum? = none ↔ xs = []

theorem List.minimum?_mem {α : Type u_1} {a : α} [Min α] (min_eq_or : ∀ (a b : α), min a b = a ∨ min a b = b) {xs : List α} : xs.minimum? = some a → a ∈ xs

theorem List.le_minimum?_iff {α : Type u_1} {a : α} [Min α] [LE α] (le_min_iff : ∀ (a b c : α), a ≤ min b c ↔ a ≤ b ∧ a ≤ c) {xs : List α} : xs.minimum? = some a → ∀ (x : α), x ≤ a ↔ ∀ (b : α), b ∈ xs → x ≤ b

theorem List.minimum?_eq_some_iff {α : Type u_1} {a : α} [Min α] [LE α] [anti : Antisymm fun (x x_1 : α) => x ≤ x_1] (le_refl : ∀ (a : α), a ≤ a) (min_eq_or : ∀ (a b : α), min a b = a ∨ min a b = b) (le_min_iff : ∀ (a b c : α), a ≤ min b c ↔ a ≤ b ∧ a ≤ c) {xs : List α} : xs.minimum? = some a ↔ a ∈ xs ∧ ∀ (b : α), b ∈ xs → a ≤ b

theorem List.minimum?_replicate {α : Type u_1} [Min α] {n : Nat} {a : α} (w : min a a = a) : (List.replicate n a).minimum? = if n = 0 then none else some a

@[simp]

theorem List.minimum?_replicate_of_pos {α : Type u_1} [Min α] {n : Nat} {a : α} (w : min a a = a) (h : 0 < n) : (List.replicate n a).minimum? = some a

maximum?

@[simp]

theorem List.maximum?_nil {α : Type u_1} [Max α] : [].maximum? = none

theorem List.maximum?_cons {α : Type u_1} {x : α} [Max α] {xs : List α} : (x :: xs).maximum? = some (List.foldl max x xs)

@[simp]

theorem List.maximum?_eq_none_iff {α : Type u_1} {xs : List α} [Max α] : xs.maximum? = none ↔ xs = []

theorem List.maximum?_mem {α : Type u_1} {a : α} [Max α] (min_eq_or : ∀ (a b : α), max a b = a ∨ max a b = b) {xs : List α} : xs.maximum? = some a → a ∈ xs

theorem List.maximum?_le_iff {α : Type u_1} {a : α} [Max α] [LE α] (max_le_iff : ∀ (a b c : α), max b c ≤ a ↔ b ≤ a ∧ c ≤ a) {xs : List α} : xs.maximum? = some a → ∀ (x : α), a ≤ x ↔ ∀ (b : α), b ∈ xs → b ≤ x

theorem List.maximum?_eq_some_iff {α : Type u_1} {a : α} [Max α] [LE α] [anti : Antisymm fun (x x_1 : α) => x ≤ x_1] (le_refl : ∀ (a : α), a ≤ a) (max_eq_or : ∀ (a b : α), max a b = a ∨ max a b = b) (max_le_iff : ∀ (a b c : α), max b c ≤ a ↔ b ≤ a ∧ c ≤ a) {xs : List α} : xs.maximum? = some a ↔ a ∈ xs ∧ ∀ (b : α), b ∈ xs → b ≤ a

theorem List.maximum?_replicate {α : Type u_1} [Max α] {n : Nat} {a : α} (w : max a a = a) : (List.replicate n a).maximum? = if n = 0 then none else some a

@[simp]

theorem List.maximum?_replicate_of_pos {α : Type u_1} [Max α] {n : Nat} {a : α} (w : max a a = a) (h : 0 < n) : (List.replicate n a).maximum? = some a